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First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
Mathematics includes the study of such topics as numbers, formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes. There is no general consensus about its exact scope or epistemological status.
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
In mathematical logic, model theory is the study of the relationship between formal theories, and their models. The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.
Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem.
Kenneth Jon Barwise was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used.
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas can be effectively determined. A theory in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership can exist for them.
Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation.
In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox, and was described as a "paradoxical state of affairs" by Skolem.
Joseph Amadee Goguen was an American computer scientist. He was professor of Computer Science at the University of California and University of Oxford, and held research positions at IBM and SRI International.
Yiannis Nicholas Moschovakis is a set theorist, descriptive set theorist, and recursion (computability) theorist, at UCLA.
Universal logic is the field of logic that studies the features common to all logical systems, aiming to be to logic what universal algebra is to algebra. A number of approaches to universal logic have been proposed since the twentieth century, using model theoretic, and categorical approaches.
In mathematical logic, Lindström's theorem states that first-order logic is the strongest logic having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.
In mathematical logic, institutional model theory generalizes a large portion of first-order model theory to an arbitrary logical system.
In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967.
The relative strength of two systems of formal logic can be defined via model theory. Specifically, a logic is said to be as strong as a logic if every elementary class in is an elementary class in .
In mathematical logic, an abstract logic is a formal system consisting of a class of sentences and a satisfaction relation with specific properties related to occurrence, expansion, isomorphism, renaming and quantification.
References
- ↑ Institution-independent model theory by Răzvan Diaconescu 2008 ISBN 3-7643-8707-6 page 3
- ↑ Handbook of mathematical logic by Jon Barwise 1989 ISBN 0-444-86388-5 page 45
- ↑ Jean-Yves Béziau Logica universalis: towards a general theory of logic 2005 ISBN 978-3-7643-7259-0 pages 20–25
- ↑ J. Barwise, 1974 "Axioms for abstract model theory", Annals of Mathematical Logic 7:221–265
